A PRESENTATION FOR A GROUP OF AUTOMORPHISMS OF A SIMPLICIAL COMPLEX - PDF document

A PRESENTATION FOR A GROUP OF AUTOMORPHISMS OF A SIMPLICIAL COMPLEX by M. A. ARMSTRONG (Received 29 May, 1987) Introduction. The Bass-Serre theorem supplies generators and relations for a group of automorphisms of a tree. Recently K. S. Brown [2] has extended the result to produce a presentation for a group of automorphisms of a simply connected complex, the extra ingredient being relations which come from the 2-cells of the complex. Suppose G is the group, K the complex and L the 1-skeleton of K. Then an extension of ni{L) by G acts on the universal covering space of L (which is of course a tree) and Brown's technique is to apply the work of Bass and Serre to this action. Our aim is to give a direct elementary proof of Brown's theorem which makes no use of covering spaces, deals with the Bass-Serre theorem as a special case and clarifies the roles played by the various generators and relations. 1. Preliminaries. By a group of automorphisms of a simplicial complex K we mean a group of homeomorphisms of the underlying polyhedron \K\ whose elements permute the simplexes of K. A directed edge e of K is a physical edge plus a choice of one of its vertices as initial vertex i(e). The second vertex is then written t(e) and called the terminal vertex. Making the other choice for i(e) produces the reverse e of e. From now on we refer to directed edges simply as edges. Let V denote the set of vertices and E the set of edges of K. These two sets together with the functions E—>E, e — * e form a graph X in the sense of Serre [3] because we clearly have e = e, e^e and i(e) = t(e) for each e e E. If G is a group of automorphisms of K we have a natural action of G on V and on E such that gi(e) = i(ge) and gt(e) = t{ge) for each g eG and e e E. We shall assume that edges of K are never reversed by the action of G. More formally, if g e G and e e E then ge ± e. Thus the quotient X/G has the structure of a graph. Because X comes from a simplicial complex the initial and terminal vertices of an edge are always different. Of course this property may well be lost on passage to X/G. A path in K (and in A^ joining vertex u to vertex v is an ordered string of edges • • e n such that /(e,) = u, i{e k+x ) = t{e k ) for l^k^n-1, and t(e n ) = v. A path of Glasgow Math. J. 30 (1988) 331-337. Downloaded from https://www.cambridge.org/core. IP address: 192.151.151.66, on 19 Aug 2020 at 20:46:10, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500007424

332 M. A. ARMSTRONG the form ee will be called a round trip. Let e x . . . e k e k+l . . . e n be a path and suppose i{e k ), t{e k ), t(e k+1 ) are the three vertices of a triangle of K. Let e be the edge of this triangle determined by i(e) = i(e k ), t(e) = t{e k+l ). Replacing the pair e k e k+l by e in our original path is called taking a short cut. We shall work with a complex which is both connected and simply connected. Each of these notions has a combinatorial description. A complex is connected if any two distinct vertices can be joined by a path. It is simply connected if two paths which join the same pair of vertices u, v are always homotopic. That is to say we can convert one path into the other (keeping a path from u to v throughout) by a finite number of steps each of which involves the addition or removal of either a round trip or a short cut. We adopt the usual notation whereby G u denotes the stabilizer of the vertex u. If g e G happens to fix u we write g u for the element g thought of as a member of G u . Of course G e denotes the stabilizer of the edge e. If u is a vertex of e then G e is a subgroup of G u . Recall that a graph is a tree if any two of its vertices may be joined by a path, and any path which joins a vertex to itself must contain a round trip. With G, K, X as above choose a maximal tree M in X/G and lift it [3, Proposition 1.14] to a subtree T of X. The vertices of T form a set of representatives for the action of G on the vertices of X. For each pair of edges/, /from X/G - M, select one, say/, and lift it to an edge e of X which has its initial vertex J C in T. Exactly one vertex z of T lies in the same orbit as t(e) and we choose an element yyfrom G that maps z onto t(e). We can now lift / to (y f )~ l e. This has its initial vertex z in T and yj = {y f )~ l sends the vertex x of T to its terminal vertex (Fig. 1). Finally we extend the correspondence /— *y f over the edges of M by setting y f = 1 (the identity element of G) whenever f eM. Figure 1 Downloaded from https://www.cambridge.org/core. IP address: 192.151.151.66, on 19 Aug 2020 at 20:46:10, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500007424

AUTOMORPHISMS OF A SIMPLICIAL COMPLEX 333 The elements of all the G w , where w is a vertex of T, and the yy, where / is an edge of XIG, together generate G. They satisfy the relation rj8*Yf = (Y- f gY f ) z whenever g fixes the canonical lift e off, plus relations which come from the triangles of K and which will be described in the next section. 2. Tail wagging. Let g eG and let e^e 2 ... e n be a path which joins a vertex v of T to gv. If the path lies entirely in T then gv = v because no two vertices of T lie in the same orbit. Therefore g = g v , where as usual g v denotes the element g interpreted as a member of G v . Otherwise there is a first edge e m that is not in T. The path e m e m+1 . . . e n will be called the tail of e t e 2 .. . e n . Let x x be the initial vertex of e m . Project e m into X/G to give an edge f x . The canonical lift e 1 of /j into X has its initial vertex in T, so i{e l ) = x r . Choose an element a Xi e G Xi which sends e 1 to e m . Let el = (y/.aj/K for m + 1 « £ k = £ n and replace e x e 2 • . . e n by the new path em+iei, +2 . . . e\. We call this process tail wagging. Our new path begins at which is a vertex of T and ends at {Yf x a~{g)v; see Fig. 2. We walk along it to the first point JC 2 where it quits T and repeat the above procedure. Since we shorten the tail at each step we eventually obtain a path which lies entirely in T and ends at, say, Then y/,aj/ . . . Yjfi^g must fix v, say y/ r aj/ . . . Y/i a 7ig = a v eG v . We now have Downloaded from https://www.cambridge.org/core. IP address: 192.151.151.66, on 19 Aug 2020 at 20:46:10, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500007424

334 M. A. ARMSTRONG This shows that the elements of the G w , w eT, together with the y f , /an edge ofX/G, do indeed generate G. Now for the extra relations mentioned in §1. Our group G acts on the collection of all triangles in K. From each orbit we choose a triangle which has a vertex in T. Let A be such a triangle and let v be a vertex of A which belongs to T. Walking round the boundary of A (there is a choice of direction here) produces a path which joins v to v = lv, and tail wagging this path expresses the identity element of G as a word r A in our generators. The missing relations have the form one for each orbit of triangles in K. With the notation established above let *G W denote the free product of the stabilizers of the vertices of T, and F the free group generated by symbols X f , one for each edge of X/G. Let R be the normal consequence in (%G W ) * F of the words X f (/ an edge of M), kjkf (for each edge of X/G), Xjg x Xf{YjgYf)~ l (when g fixes the canonical lift e of/), and r^ (obtained from r A by changing each occurrence of y f to Xf, one such for each orbit of triangles). PRESENTATION THEOREM. If G is a group of automorphisms of a connected simply connected complex K, and if no edge of K is reversed by the action of G, then G is isomorphic to [(*G W ) * F]/R. If K is one-dimensional, so that A' is a tree, this is the Bass-Serre Theorem [3, Theoreme 1.13]. For dimension two or more the extra relations were provided by K. S. Brown [2]. 3. Homotopy. We shall produce an isomorphism as follows. Take a vertex v of T as base point. Given g e G, choose a path a in K which joins v to gv. By tail wagging a we express g as a word a Xl y fl .. . a Xr Yf r 0v and we define Of course various choices are involved here and we must show that ty is well defined. For a particular path a joining v to gv the first ambiguity occurs in the choice of the element a Xi e G Xt which maps e 1 to e m . That a different choice at this and subsequent stages gives the same coset for ip(g) is verified exactly as in [1]. As to the choice of the actual path a we need only check that altering a by the addition or removal of a single round trip or short cut makes no difference to the value of ip(g). Downloaded from https://www.cambridge.org/core. IP address: 192.151.151.66, on 19 Aug 2020 at 20:46:10, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500007424